Abstract: We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for $n$-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.

Abstract: This work develops compressed sensing strategies for computing the dynamic mode decomposition (DMD) from heavily subsampled or compressed data.
The resulting DMD eigenvalues are equal to DMD eigenvalues from the full-state data.
It is then possible to reconstruct full-state DMD eigenvectors using $\ell_1$-minimization or greedy algorithms.
If full-state snapshots are available, it may be computationally beneficial to compress the data, compute DMD on the compressed data, and then reconstruct full-state modes by applying the compressed DMD transforms to full-state snapshots.
These results rely on a number of theoretical advances. First, we establish connections between DMD on full-state and compressed data. Next, we demonstrate the invariance of the DMD algorithm to left and right unitary transformations.
When data and modes are sparse in some transform basis, we show a similar invariance of DMD to measurement matrices that satisfy the restricted isometry property from compressed sensing.
We demonstrate the success of this architecture on two model systems.
In the first example, we construct a spatial signal from a sparse vector of Fourier coefficients with a linear dynamical system driving the coefficients.
In the second example, we consider the double gyre flow field, which is a model for chaotic mixing in the ocean.

Abstract: Optimal control problems for underactuated
mechanical systems can be seen as a higher-order variational problem
subject to higher-order constraints (that is, when the Lagrangian
function and the constraints depend on higher-order derivatives such
as the acceleration, jerk or jounces). In this paper we discuss the
variational formalism for the class of underactuated mechanical
control systems when the configuration space is a trivial principal
bundle and the construction of variational integrators for such
mechanical control systems.
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An interesting family of geometric integrators can be defined using
discretizations of the Hamilton's principle of critical action. This
family of geometric integrators is called variational integrators,
being one of their main properties the preservation of geometric
features as the symplecticity, momentum preservation and good
behavior of the energy. We construct variational
integrators for higher-order mechanical systems on trivial principal
bundles and their extension for higher-order constrained systems, paying particular attention to the case of underactuated
mechanical systems.

Abstract: We present a numerical technique for the computation of a Lyapunov function
for nonlinear systems with an asymptotically stable equilibrium point. The proposed approach
constructs a partition of the state space, called a triangulation, and then computes values at
the vertices of the triangulation using a Lyapunov function from a classical converse Lyapunov
theorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuous
and Piecewise Affine (CPA) function.
Verification that the obtained CPA function is a Lyapunov function is shown
to be equivalent to verification of several simple inequalities.
Numerical examples are presented demonstrating different aspects of the proposed method.

Abstract: A data-driven, kernel-based method for approximating the leading Koopman
eigenvalues, eigenfunctions, and modes in problems with high-dimensional state
spaces is presented.
This approach uses a set of scalar observables (functions that map a state to a scalar value) that are defined implicitly by the feature map associated with a user-defined kernel function.
This circumvents the computational issues that arise due to the number of
functions required to span a ``sufficiently rich'' subspace of all possible scalar observables in such applications.
We illustrate this method on two examples: the first is the FitzHugh-Nagumo PDE,
a prototypical one-dimensional reaction-diffusion
system, and the second is a set of vorticity data computed from experimentally obtained
velocity data from flow past a cylinder at Reynolds number 413.
In both examples, we use the output of Dynamic Mode Decomposition, which has a similar computational cost, as the benchmark for our approach.